The On-Line Encyclopedia of Integer Sequences - Neil Sloane

The On-Line Encyclopedia of Integer Sequences Neil J. A. Sloane Visiting Scholar, Rutgers University President, OEIS Foundation 11 South Adelaide Avenue Highland Park, NJ

Monday, March 16, 15

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The On-Line Encyclopedia of Integer Sequences® (OEIS®) Enter a sequence, word, or sequence number: 1,2,3,6,11,23,47,106,235 Search

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Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages The OEIS Community | Maintained by The OEIS Foundation Inc. Content is available under The OEIS End-User License Agreement . Last modified March 16 18:25 EDT 2015. Contains 255620 sequences. Monday, March 16, 15

Lunar


Outline of Talk • About the OEIS • Sequences from Geometry • Sequences from Arithmetic • “Music” and Videos • The BANFF “Integer Sequences and K12” Conference


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• Fun: 2, 4, 6, 3, 9, 12, 8, 10, 5, 15, ... ? • Addictive (better than video games) • Accessible (free, friendly) • Street creds (4000 citations) • Interesting, educational • Essential reference • Low-hanging fruit • Need editors Monday, March 16, 15

Facts about the OEIS • • • • • • • Monday, March 16, 15

Accurate information about 250000 sequences Definition, formulas, references, links, programs View as list, table, graph, music! 200 new entries and updates every day 4000 articles and books cite the OEIS Often called one of best math sites on the Web Since 2010, a moderated Wiki

Main Uses for OEIS • To see if your sequence is new, to find references, formulas, programs

• Catalan or Collatz? • Many collaborations, very international • Source of fascinating research problems • Has led many people into mathematics • Fun, Escape (Very easy or very hard?)


Sequences from Geometry • Slicing a pancake • Kobon triangles • Lines in plane • Circles in plane •


Tiling a square with dominoes

A124

a(n)=a(n-1)+n;


a(n) = n(n+1)/2 + 1

Kobon Triangles Kobon Fujimura, circa 1983 A6066


Kobon Triangles How many non-overlapping triangles can you draw with n lines? 1 2 3 4 5 6 7 8 9 0 0 1 2 5 7 11 15 21

A6066 a(5)=5 a(10) is 25 or 26


Kobon triangles (cont.)

a(17)=85 Monday, March 16, 15

A6066

Johannes Bader: 17 lines, 85 triangle, maximum possible

No. of ways to arrange n lines in the plane 1, 2, 4, 9, 47, 791, 37830 A241600


A241600 (cont.)


A90338 A subset: n lines in general position 1,1,1,1, 6, 43, 922, 38609 Wild and Reeves, 2004

a(5)=6 5 lines in general position: 6 ways


A250001 No. of arrangements of n circles in the plane 1, 3, 14, 168 Jonathan Wild What if allow tangencies?


Tiling a Square with Dominoes 36 ways to tile a 4X4 square

a(2)=36

1, 2, 36, 6728, 12988816, 258584046368, 53060477521960000 ... (A4003) ,

a(n) =

n Y n ✓ Y

j=1 k=1 Monday, March 16, 15

j⇡ k⇡ 2 4 cos + 4 cos 2n + 1 2n + 1 2

◆

(Kastelyn, 1961)

Two days ago: Laura Florescu, Daniela Morar, David Perkinson, Nicholas Salter, Tianyuan Xu, of 2 divides a . Dominos, We do not know when equality holds, and we have not yet answered Sandpiles and 2015 Irena Swanson’s question. 2n 2n

n

On the other hand, further experimentation using the mathematical software Sage led us to a more fundamental connection between the sandpile group and domino tilings of the grid graph. The connection is due to a property that is a notable feature of the elements of the subgroup generated by the all-2s configuration—symmetry with respect to the central horizontal and vertical axes. The recurrent identity element for the sandpile grid graph, as exhibited in Figure 1, also has this symmetry.3

1, 2, 36, 6728, 12988816, 258584046368, 53060477521960000/5, ... !! (A256043) # grains =0 =1 =2 =3

Figure 1: Identity element for the sandpile group of the 400


400 sandpile grid graph.

If is any graph equipped with an action of a finite group G, it is natural to consider the collection of G-invariant configurations. Proposition 6 establishes that the symmetric

Two Sequences That Agree For a Long Time

l

j 2n k log 2 2

21/n

1

= A078608 m

Differs for first time at n = 777451915729368 (see A129935)


Sequences from Arithmetic

• Lunar primes • The EKG sequence • Curling number conjecture • Gijswijt’s sequence


Lunar Arithmetic David Applegate, Marc LeBrun and NJAS ( J.I.S. 2011) (For Martin Gardner)

Arithmetic on the moon!

• • •

i + j = MAX {i, j} i × j = MIN {i, j} No carries! 17 + 23 ----27


19 × 24 -----14 12 -----124

Thm.: Lunar arithmetic is commutative, associative, distributive

Lunar squares 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 100, 111, 112, 113, 114, 115, 116, 117, 118, 119, 200, ... (A87019)

Lunar primes? Monday, March 16, 15

19 ×19 ----19 110 ----119

What is a prime? Ans. Only factorization is p = 1 × p But what is “1”? Ans. The multiplicative identity: “1” × n = n for all n But 1 × 3 = 1, so “1” ≠ 1 So “1” = 9, since 9 × n = n for all n. If u × v = 9 then u = v = 9, so 9 is the only unit.

So p is prime if its only factorization is p = 9 × p Monday, March 16, 15

Lunar primes (cont.) p is prime if only factorization is p = 9 × p 7? No, 7 = 7 × 7 13? No, 13 × 4 So must have 9 as a digit. 9? No, 9 is the unit Lunar primes:

(A87097) 19, 29, 39, 49, 59, 69, 79, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 109, 209, ... (119 = 19 × 19 is not a prime)


There are infinitely many primes 109 is prime! Proof: ab × cd

19 c9 ------19 1c0 ------1?9

Can’t get 109

Similarly, 10...009 is prime!


The EKG Sequence Jonathan Ayres, 2001


EKG Sequence (A64413) 1, 2, 4, 6, 3, 9, 12, 8, 10, 5, 15, ... a(1)=1, a(2)=2, a(n) = min k such that • GCD { a(n-1), k } > 1 • k not already in sequence - Jonathan Ayres, 2001 - Analyzed by Lagarias, Rains, NJAS, Exper. Math., 2002 - Gordon Hamilton,Videos related to this sequence: https://www.youtube.com/watch?v=yd2jr30K2R4&feature=youtu.be


http://m.youtube.com/watch?v=Y2KhEW9CSOA

EKG Sequence


EKG Sequence


EKG Sequence Normally a(n) ≈ n, but if a(n) = prime then a(n) ≈ n/2, if a(n) =3 × prime then a(n) ≈ 3n/2

Text


EKG Sequence

Theorems:

{a(n)} is a ispermutation {1, .natural . . n} numbers sequence a permutationofof the • The c1 n  a(n)  c2 n

•

Conjectures • •

✓

◆

1 a(n) ⇠ n 1 + for the main terms 3 log n · · · , 2p, p, 3p, · · · (primes p > 2)

(Proved by Hofman & Pilipczuk, 2008)


EKG Sequence


The Curling Number Conjecture


The Curling Number Conjecture Definition Y of Y Curling ... Number Y

S X

S = 7 5 2 2 5 2 2 5 2 2, k = 3 Monday, March 16, 15


Gijswijt’s Sequence Dion Gijswijt, 2004 A90822


Gijswijt’s Sequence Fokko v. d. Bult, Dion Gijswijt, John Linderman, N. J. A. Sloane, Allan Wilks (J. Integer Seqs., 2007) Start with 1, always append curling number 1

1

2

1

1

2

1

1

2

1

1

2

1

1

2

1

1

2

1

1

2

1

1

2

1

1

2

.

.

.

a(220) = 4 Monday, March 16, 15

2

2

3

a(20) = 4

a(20) = a(220) = 44 2

2

3

2

2

3

2

2

3

.

.

.

2

2

2

2

3

2

2

2

3

3

2

(A090822)

Gijswijt, continued


Is there a 5?


Gijswijt, continued

Is there a 5? 300,000 terms: no 5


Gijswijt, continued


2 · 10 terms: no 5 6


Gijswijt, continued


2 · 10 terms: no 5 6

10

120


terms: no 5

Gijswijt, continued

Is there a 5?

Gijswijt, continued

300,000 terms: no 5

2 · 10 terms: no 5 6

10

120

terms: no 5

23 10 NJAS, FvdB: first 5 at about term 10


Gijswijt, continued

First n appears at about term n

1

... 5 4 3 2 2 (F.v.d. Bult et al., J. Integer Sequences, 2007) Monday, March 16, 15

(A90822)

Gijswijt, continued

Proofs could be simplified if Curling Number Conjecture were true How far can you get with an initial string of n 2’s and 3’s (before a 1 appears)?



Conjecture

Curling Number Conjecture, continued

Searched n